Definition Coherent nerve of pseudo double category [efr-001S]

Let \mathbb {C} be a pseudo double category. Then the coherent nerve is the simplicial category N(\mathbb {C}) defined as follows:

  1. N(\mathbb {C})_0 = \mathbb {C}_0
  2. N(\mathbb {C})_1 = \mathbb {C}_1
  3. The objects of N(\mathbb {C})_2 are tuples f: x \nrightarrow y, g: y \nrightarrow z, h: x \nrightarrow z, \alpha : gf \simeq h, where x,y,z are objects of \mathbb {C}_0, f,g,h are horizontal morphisms, and \alpha is a globular isomorphism in \mathbb {C}_1
  4. The morphisms of N(\mathbb {C})_2 are triples of 2-cells \phi : f \to f', \psi : g \to g', \xi : h \to h' so that the two possible 2-cells gf \to h' agree
  5. Given four compatible objects of \N (\mathbb {C})_2 defining a 3-boundary (a tetrahedron), there is a unique objects of \N (\mathbb {C})_3 filling it if and only if the square of globular isomorphisms commute
  6. Given four compatible morphisms of \N (\mathbb {C})_2, there is always a unique morphism in \N (\mathbb {C})_3 filling the boundary
  7. The face- and degeneracy functors are the obvious ones
  8. \N (\mathbb {C}) is 3-coskeletal, which defines the rest of the structure